Solve each triangle ABC.

B = 38° 40', a = 19.7 cm, C = 91° 40'

Video transcript

Hey, everyone here we are asked to find the missing side links and angle of the triangle ABC, we are given capital B is equal to 41 degrees and 30 minutes. Lowercase A is equal to 23.5 centimeters and capital C is equal to 97 degrees and 20 minutes. We have four answer choice options, answer choices A through D, each of which state some angle for capital A as well as varying lengths for lowercase B and lowercase C. So to begin this problem, we want to first focus on finding our missing angle. So first, we need to recall the angle some property of a triangle which for triangle ABC states that angle A plus angle B plus angle C is equal to 180 degrees. Now, just substituting in our given information, we have unknown angle, A plus 41 degrees and 30 minutes plus 97 degrees and 20 minutes is equal to 180 degrees. Now just summing the terms on the left hand side, we have angle A plus 138 degrees and 50 minutes is equal to 180 degrees. Next, we want to recall that there are 60 minutes in one degree. So now just solving for our angle A, we find that angle of capital A here is equal to 41 degrees and 10 minutes. And so now that we have found our missing angle, we can move on to, to find our missing side links. So for this, we need to recall the law of signs which for our triangle ABC states that lowercase A divided by S of capital A is equal to lowercase B divided by S of capital B, which is also equal to lower case C divided by sine of capital C. Looking at our formula, we can notice that we already have our lower case A and capital A. So we can go ahead and utilize the A expression to solve for our other side links. So first solving for B, we again want to use the A expression from the law of signs. And so just substituting in our given information here, we have 23.5 divided by S of capital A which we found to be 41 degrees and 10 minutes. And now from the law of science, we know this expression is equal to lowercase B divided by sign of capital B which is given as 41 degrees and 30 minutes. And so now we just want to solve this equation for B by just multiplying both sides by the denominator on the right hand side. And so now we find that B is equal to 23.5 multiplied by sine of 41 degrees and 30 minutes. And this is all divided by s of 41 degrees and 10 minutes. And finally, just evaluating this expression here, we find that B is equal to 23.66 centimeters. And now that we have found one missing side length, we can find our second missing length by again, utilizing this a expression. So now just substituting again, our given information, we have 23.5 divided by sign of 41 degrees and 10 minutes. And we know that from the law of signs, this is equal to C divided by sign of 97 degrees and 20 minutes. Now just solving this equation for C, just like we did with B, we have that C is equal to 23.5 multiplied by sine of 97 degrees and 20 minutes all divided by s of 41 degrees and 10 minutes. Now, evaluating our final expression here, we find that C is equal to 35.41 centimeters. And with this final value, we have finished finding our missing angle as well as both of our missing side lengths. And so we are left with answer choice B where again, we found that capital A or angle A is equal to 41 degrees and 10 minutes lowercase B is equal to 23.66 centimeters and lowercase C is equal to 35.41 centimeters. Thanks for watching. I hope you found this video helpful and I'll see you again next time.